Revert before introduction of option Assert
This mostly reverts commit c2c58119be.
The theorems will be slightly different and should be way cleaner.
This commit is contained in:
parent
260ac05c6e
commit
cd467dc8b0
1 changed files with 34 additions and 90 deletions
124
wp.v
124
wp.v
|
|
@ -476,98 +476,42 @@ Proof.
|
|||
intros F; exfalso; exact F.
|
||||
Qed.
|
||||
|
||||
Theorem wp_correctness (instr: Instr):
|
||||
forall post, ( |= [| wp instr post |] instr [| post |] ) % assert.
|
||||
Theorem wp_correctness_provable (instr: Instr) :
|
||||
forall post, ( |- [| wp instr post |] instr [| post |] ) % assert.
|
||||
Proof.
|
||||
induction instr; intros post.
|
||||
* apply hoare_provability_implies_consequence.
|
||||
apply (H_skip post).
|
||||
* apply hoare_provability_implies_consequence.
|
||||
apply (H_abort assertTop post).
|
||||
* apply hoare_provability_implies_consequence. apply (H_assign post v e).
|
||||
* apply hoare_provability_implies_consequence.
|
||||
remember (wp instr2 (Some post)) as mid eqn:midRel.
|
||||
induction instr; intros post; simpl.
|
||||
* apply (H_skip post).
|
||||
* apply (H_abort assertTop post).
|
||||
* apply (H_assign post v e).
|
||||
* remember (wp instr2 post) as mid eqn:midRel.
|
||||
remember (wp instr1 mid) as pre eqn:preRel.
|
||||
simpl; rewrite <- midRel; rewrite <- preRel.
|
||||
specialize IHinstr2 with (Some post0) as IHpost.
|
||||
specialize IHinstr1 with mid as IHmid.
|
||||
rewrite <- midRel in IHpost; rewrite <- preRel in IHmid.
|
||||
destruct mid as [mid0 | ] eqn:mid0Rel.
|
||||
destruct pre as [pre0 | ] eqn: pre0Rel.
|
||||
apply (H_seq pre0 mid0 post0).
|
||||
apply IHmid. apply IHpost.
|
||||
+ unfold whatever_or_none; trivial.
|
||||
+ destruct pre.
|
||||
- unfold provable_or_none in IHmid; unfold whatever_or_none in IHmid.
|
||||
exfalso. apply IHmid.
|
||||
- unfold whatever_or_none; trivial.
|
||||
* apply hoare_provability_implies_consequence.
|
||||
specialize IHinstr1 with (Some post0);
|
||||
specialize IHinstr2 with (Some post0).
|
||||
destruct (wp instr1 (Some post0)) as [preIf | ] eqn:preIfRel;
|
||||
destruct (wp instr2 (Some post0)) as [preElse | ] eqn:preElseRel.
|
||||
remember ((assertOfExpr e -> preIf)
|
||||
/\ (~ assertOfExpr e -> preElse)) % assert
|
||||
as pre0 eqn:pre0Rel.
|
||||
assert (wp (ifelse e instr1 instr2) (Some post0) = (Some pre0)).
|
||||
{
|
||||
rewrite pre0Rel; simpl; rewrite preIfRel; rewrite preElseRel; congruence.
|
||||
}
|
||||
{
|
||||
rewrite H.
|
||||
apply (H_if pre0 post0 e instr1 instr2).
|
||||
- apply (H_conseq
|
||||
(pre0 /\ assertOfExpr e)%assert post0
|
||||
preIf post0 instr1
|
||||
IHinstr1).
|
||||
+ rewrite pre0Rel. unfold assertImplLogical.
|
||||
intros mem [ [disjunctIf disjunctElse] isIf].
|
||||
apply (assertImplElim mem disjunctIf isIf).
|
||||
+ apply (assertImplSelf post0).
|
||||
- apply (H_conseq
|
||||
(pre0 /\ ~ assertOfExpr e)%assert post0
|
||||
preElse post0 instr2
|
||||
IHinstr2).
|
||||
+ rewrite pre0Rel. unfold assertImplLogical.
|
||||
intros mem [ [disjunctIf disjunctElse] isElse].
|
||||
apply (assertImplElim mem disjunctElse isElse).
|
||||
+ apply (assertImplSelf post0).
|
||||
}
|
||||
|
||||
- unfold provable_or_none; simpl; rewrite preElseRel;
|
||||
rewrite preIfRel; trivial.
|
||||
- unfold provable_or_none; simpl; rewrite preElseRel;
|
||||
rewrite preIfRel; trivial.
|
||||
- unfold provable_or_none; simpl; rewrite preElseRel;
|
||||
rewrite preIfRel; trivial.
|
||||
* unfold wp. apply preBottomIsCorrect.
|
||||
Qed.
|
||||
|
||||
Lemma provable_opt_implies_provable {pre instr post} :
|
||||
(|-opt [|Some pre|] instr [|Some post|])%assert
|
||||
-> (|- [|pre|] instr [|post|])%assert.
|
||||
Proof.
|
||||
intros prf.
|
||||
unfold provable_or_none in prf; unfold whatever_or_none in prf. assumption.
|
||||
Qed.
|
||||
|
||||
Theorem wp_correctness (instr: Instr) :
|
||||
forall post, ( |=opt [| wp instr post |] instr [| post |] ) % assert.
|
||||
Proof.
|
||||
intros post.
|
||||
destruct post as [post0 | ] eqn:post0Rel.
|
||||
remember (wp instr (Some post0)) as pre eqn:preRel.
|
||||
destruct pre as [ pre0 | ] eqn:pre0Rel.
|
||||
- unfold consequence_or_none; unfold whatever_or_none.
|
||||
apply hoare_provability_implies_consequence.
|
||||
apply provable_opt_implies_provable.
|
||||
rewrite preRel.
|
||||
apply wp_correctness_provable.
|
||||
- unfold consequence_or_none; unfold whatever_or_none; trivial.
|
||||
- unfold wp.
|
||||
destruct instr; unfold consequence_or_none; unfold
|
||||
whatever_or_none; trivial.
|
||||
Qed.
|
||||
specialize IHinstr2 with post.
|
||||
specialize IHinstr1 with mid.
|
||||
rewrite <- midRel in IHinstr2.
|
||||
rewrite <- preRel in IHinstr1.
|
||||
apply (H_seq pre mid post instr1 instr2).
|
||||
assumption. assumption.
|
||||
* remember ((assertOfExpr e -> wp instr1 post)
|
||||
/\ (~ assertOfExpr e -> wp instr2 post)) % assert
|
||||
as pre eqn:preRel.
|
||||
apply (H_if pre post e instr1 instr2).
|
||||
- apply (H_conseq
|
||||
(pre /\ assertOfExpr e)%assert post
|
||||
(wp instr1 post) post instr1
|
||||
(IHinstr1 post)).
|
||||
+ rewrite preRel. unfold assertImplLogical.
|
||||
intros mem. intros [ [disjunctIf disjunctElse] isIf].
|
||||
apply (assertImplElim mem disjunctIf isIf).
|
||||
+ apply (assertImplSelf post).
|
||||
- apply (H_conseq
|
||||
(pre /\ ~ assertOfExpr e)%assert post
|
||||
(wp instr2 post) post instr2
|
||||
(IHinstr2 post)).
|
||||
+ rewrite preRel. unfold assertImplLogical.
|
||||
intros mem. intros [ [disjunctIf disjunctElse] isElse].
|
||||
apply (assertImplElim mem disjunctElse isElse).
|
||||
+ apply (assertImplSelf post).
|
||||
*
|
||||
|
||||
(* vim: ts=2 sw=2
|
||||
*)
|
||||
|
|
|
|||
Loading…
Reference in a new issue